17th Internet Seminar

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The 17th Internet Seminar on Evolution Equations is devoted to positive linear dynamical systems. Motivated by numerous applications in life sciences, we present an operator theoretical treatment to study quantitative and qualitative properties of positive semigroups both in finite and infinite dimension.

The lectures are at a beginning graduate level and assume basic familiarity with linear algebra, functional analysis as well as with ordinary and partial differential equations.

Organised by the European consortium "International School on Evolution Equations", the annual Internet Seminars introduce master, Ph.D. and postdoc students to varying subjects related to evolution equations. The course consists of three phases.

  • In Phase 1 (October-February), a weekly lecture will be provided via the ISEM website. Our aim is to give a thorough introduction to the field, at a speed suitable for master's or Ph.D. students. The weekly lecture will be accompanied by exercises, and the participants are supposed to solve these problems.
  • In Phase 2 (March-May), the participants will form small international groups to work on diverse projects which supplement the theory of Phase 1 and provide some applications of it.
  • Finally, Phase 3 (22-28 June 2014) consists of the final one-week workshop at the Heinrich--Fabri Institut in Blaubeuren (Germany). There the teams will present their projects and additional lectures will be delivered by leading experts.

ISEM team 2013/14 :

Virtual lecturers
András Bátkai (Budapest)
Marjeta Kramar Fijavž (Ljubljana)
Abdelaziz Rhandi (Salerno)
Abdelaziz Rhandi
Rosanna Manzo
Cristian Tacelli

Description of the course

The aim of the course is to study positivity and spectral properties of the solution to the initial value Cauchy problem

u'(t) &=Au(t),\quad t\geq 0,\\
u(0) &=u_0\in D(A),

where A generates a C_0-semigroup (T(t))_{t\ge 0} on a Banach lattice X, and u_0 is the positive initial value. The main body of this course is the finite dimensional case, i.e. when X={\mathbb R}^N and A=(a_{ij}) is a real constant N\times N matrix. In this case, it can be seen that the solution to the above problem is given by u(t)=T(t)u_0=e^{tA}u_0 and the matrix e^{tA} is positive for all t\ge 0 if and only if A-{\rm diag}(A)\ge 0, that is, a_{ij}\ge 0 for i\neq j. For such matrices, Perron (1907) and Frobenius (1909) discovered remarkable spectral properties which determine the asymptotic behavior of the solutions to the above Cauchy problem.

In infinite dimension, W. Feller (1952) and R.S. Phillips (1962) obtained first results concerning the characterization of the generators of positive semigroups. Thanks to the development of the theory of ordered Banach spaces and positive operators in the 60's and 70's, many applications of positivity to concrete evolution equations from transport theory, mathematical biology, and physics have been studied. Most results of what was known around 1985 about this subject can be found in the monograph written by the functional analysis group in Tübingen [1]. This led to further progress during the last decades.

Topics to be covered include:

Part I Finite dimension
  • Functional calculus and matrix exponential function,
  • positive matrices and the Perron-Frobenius theory,
  • applications: graph theory, control theory, etc.

Part II Infinite dimension
  • An introduction to Banach lattices, positive operators and positive semigroups,
  • spectral theory for positive semigroups,
  • positive semigroups for transport and population equations.

More applications will be elaborated on in Phase 2, where the students will have the possibility to work on projects related to active research.

You can download the poster and the announcement for the Internet Seminar here: Poster.pdf Announcement.pdf


  1. R. Nagel (ed.), One-parameter Semigroups of Positive Operators, Lect. Notes in Math., vol. 1184, Springer-Verlag, 1986.